Market Structure and Competition

Besanko and Braeutigam, CH 13

Hans Martinez

Western University

Chapter 13 Overview

  • What happens when there is more than one seller, but still there are few enough to affect the market?
  • How do their choices affect each other?
  • Will they cooperate? Will they compete?
  • What happens if their products are imperfect substitutes?

Objectives

  • Describing and Measuring Market Structure
  • Oligopoly with Homogeneous Products
  • Dominant Firm Markets
  • Oligopoly with Horizontally Differentiated Products
  • Monopolistic Competition
  • The Cournot Equilibrium and the Inverse Elasticity Pricing Rule

Market Structures: 2 Key Dimensions

Market Structures Characteristics
Market Structure Number of Firms Type of Product Control Over Price Examples
Perfect Competition Many Homogeneous None Agriculture (US)
Monopolistic Competition Many Differentiated Some Retail stores
Oligopoly Few Homogeneous or Differentiated Some to Significant Banking-Big 5 (CA)
Monopoly One Unique Significant Utilities
Dominant Firm One dominant, Many small Homogeneous Significant by dominant firm US: Scotch Tape (3M)

Measures of Market Structure

  • Four-Firm Concentration Ratio (4CR): The share of industry sales revenue accounted for by the four firms with the largest sales revenue in the industry

  • Herfindahl–Hirschman Index (HHI): The sum of the squares of the market share of each firm in the industry (\(0 \le HHI \le 10,000\))

  • Market Structure 4CR HHI
    Perfect Competition Low Low
    Olygopoly Intermediate Intermediate
    Monopoly 100 10,000

Oligopoly with Homogeneous Goods

  • A central feature of oligopoly markets: competitive interdependence
    • The decisions of every firm significantly affect the profits of competitors
    • Perfect Competition: No impact of one firm on its rivals
    • Monopolist: No rivals
  • Central question of oligopoly theory: how does the close interdependence among firms in the market affect their behavior?

The Cournot Model

  • 2 firms (duopoly); one homogeneous good
  • Both with identical marginal costs
  • Firms choose output (how much to produce)
    • simultaneously
    • non-cooperatively (no collusion)
    • no knowledge of each other’s plan

The Cournot Model

  • Inverse demand is downward-slopping and a function of the combined output of the two firms, \(P(Q_1+Q_2)\)
    • Price is not known until both firms have made their output choice
  • Each firm will produce the output choice that maximizes its profit based on its expectation of the other firm’s output choice

Residual Demand

  • Suppose firm 1 expects that firm 2 will produce \(Q^e_2\)
  • If firm 1 produces \(Q_1\), then total output will be \(Q=Q_1+Q_2^e\), and
  • and market price will be \(P(Q)=P(Q_1+Q_2^e)\)
  • the inverse market demand resulting from holding their rivals’ output constant is the residual demand
    • Ex. Linear Demand \[ P(Q)=a-b(Q_1+Q_2^e)=(a-bQ_2^e)-bQ_1 \]

Residual Demand

Example

  • Inverse demand curve \(P(Q)=100-Q\)
  • \(MC=10\)
  • Firm 1 is Samsung and Firm 2 is SK
  • If \(Q=80\), then \(P(80)=20\)
  • If \(Q^e_2=50\), then \(P(Q_1+50)=(100-50)-Q_1\)
    • for \(Q_1=30\), \(P(30+50)=20\)

Residual Demand

Best Response

  • When choosing output, each firm will act as a monopolist relative to its residual demand

  • Firm 1 maximizes its profit by choosing \(Q_1\) that maximizes: \[ \pi_1=P(Q_1+Q_2^e)⋅Q_1−C(Q_1) \]

  • Best Response: For any given belief about the output of firm 2, \(Q^e_2\), there is an optimal choice of output for firm 1, \[ Q_1=BR_1(Q^e_2) \]

Reaction Function

  • The best response function for Firm 1, \(BR_1(Q_2^e)\), is derived by setting: \[ \frac{\partial \pi_1}{\partial Q_1} = 0 \]

  • Example with Linear Demand: Assuming \(P(Q) = a - b(Q_1 + Q_2)\) and constant marginal cost \(MC = c\), the reaction functions can be simplified as: \[ \begin{aligned} Q_1= \frac{a - c}{2b} - \frac{Q_2^e}{2} \;;\; Q_2= \frac{a - c}{2b} - \frac{Q_1^e}{2} \end{aligned} \]

Reaction Function

Example (continued)

  • With \(P(Q)=100-Q\) and \(MC=10\)
  • \[ Q_1=45-\frac{Q^e_2}{2} \;;\;Q_2=45-\frac{Q^e_1}{2} \]

Rection Function

  • \(Q_1(Q^e_2)=45-\frac{Q^e_2}{2}\)
  • \(Q_1(50)=20\)
  • \(Q_1(30)=30\)
  • \(Q_1(20)=35\)

Reaction Function: Illustrates graphically a firm’s best response output for each possible output of the other firm’s output

Cournot Equilibrium

  • Cournot Equilibrium: each firm maximizes its profits, given its beliefs about the other firm’s output choice, and those beliefs are confirmed in equilibrium
    • Each firm optimally produces the output its rival expects it to produce
    • No firm has incentives to deviate
  • Mathematically, a combination of output choices \((Q_1^*,Q_2^*)\) that satisfy \[ \begin{aligned} Q^*_1= \frac{a - c}{2b} - \frac{Q_2^*}{2} \;;\; Q^*_2= \frac{a - c}{2b} - \frac{Q_1^*}{2} \end{aligned} \]

Cournot Equilibrium

  • Cournot Equilibrium for Two Firms: Occurs where the reaction curves of Firm 1 and Firm 2 intersect.

  • Equilibrium Output: Solving the system of equations given by the reaction functions yields the equilibrium outputs for both firms: \[ Q_1^* = Q_2^* = \frac{a - c}{3b} \] and the total market output is: \[ Q^* = Q_1^* + Q_2^* = \frac{2(a - c)}{3b} \]

Example (continued)

  • Continuing with our example \(P=100-Q\) and \(MC=10\)

  • Firm output is \(Q_i=30\), total market output is \(Q=60\)

  • So, market price is \(P=40\)

Achieving Equilibrium

In the Cournot equilibrium both firms fully understand their interdependence and have confidence in each other’s rationality

Cournot for N-firms

  • For \(N\) identical firms, total output is \(Q=\sum_i^N Q_i\)

  • Firm \(i\) solves \[ \max_{Q_i} P(Q)⋅Q_i-C(Q_i) \qquad(1)\]

  • For linear demand, and \(MC=c\), each firm \(i\) will produce \[ Q^*_i = \frac{1}{N+1}\frac{a - c}{b} \]

Cournot for N-firms

  • Total market output and market price will be \[ Q^*=\sum_i Q^*_i = \frac{N}{N+1}\frac{(a - c)}{b} \] \[ P^*=a-b\left(\frac{N}{N+1}\frac{(a - c)}{b} \right) = \frac{a+Nc}{(N+1)} \]

Comparing Market Structures

Market Structure \(N\) Optimal Firm Output \(Q^*_i\) Total Market Output \(Q^*\) Market Price \(P^*\)
Monopoly 1 \(\frac{a - c}{2b}\) \(\frac{a - c}{2b}\) \(\frac{a + c}{2}\)
Cournot Duopoly 2 \(\frac{a - c}{3b}\) \(\frac{2(a - c)}{3b}\) \(\frac{(a + 2c)}{3}\)
Perfect Competition \(\infty\) 0 (virtually) \(\frac{a - c}{b}\) \(c\)

Comparing Market Structures

  • Monopoly: A single firm controls the entire market, producing half the competitive output and selling at a higher price.
  • Cournot Duopoly: Two firms share the market, leading to increased output and lower prices compared to monopoly.
  • Perfect Competition: An infinite number of firms produce at marginal cost, leading to the highest output and lowest price.

Comparing Market Structures

Cournot vs Monopoly

By independently maximizing their own profits, firms produce more total output than they would if they collusively maximized industry profits (Monopoly).

Cournot and Elasticity of Demand

  • FOC of Equation 1 gives \[ P(Q)+\frac{\partial P}{\partial Q}⋅Q_i=MC(Q_i) \]

  • It can be shown that \[ P(Q)\left[1-\frac{1}{|\epsilon_{Q,P}|/s_i}\right]=MC(Q_i) \] where \(s_i=Q_i/Q\) is the market share

Cournot and Elasticity of Demand

  • \(|\epsilon_{Q,P}|/s_i\), the elasticity of the demand curve facing the firm:
    • the smaller the market share of the firm, the more elastic the demand curve it faces
  • \(s_i\) Demand Curve Condition Market Structure
    0 Flat \(P=MC\) Perfect Competition
    1 Market Demand \(P(1-\frac{1}{|\epsilon_{Q,P}|})=MC\) Monopoly

Bertrand Competition

Simultaneous Price Setting

  • Price Competition: Unlike Cournot, where firms compete by choosing quantities, in the Bertrand model, firms compete by setting prices and letting the market determine the quantity sold. Simultaneous, non-cooperative.
  • Homogeneous Products: Firms produce identical products, making price the sole factor for consumers when choosing between firms
  • Cost Structures: Firms have identical cost structures and sufficient capacity to meet all demand
  • Rational Expectations: Firms are profit-maximizers and have rational expectations about their competitors’ behavior

Bertrand Model

  • Profit Maximization for Firm 1: \[ \pi_1 = P_1 \cdot Q_1(P_1, P_2) -C(Q_1(P_1,P_2)) \]
    • where \(P_1\) is the price set by Firm 1,
    • \(C(Q_1(P_1,P_2))\) is cost function for Firm 1, and
    • \(Q_1(P_1, P_2)\) is the demand for Firm 1’s product as a function of both firms’ prices

Demand Scenarios for Firm 1

  1. When \(P_1 = P_2\):
    • If both firms set equal prices, we assume demand is split equally between them. Hence, \(Q_1 =Q_2 = \frac{1}{2}Q(P_2)\), where \(Q(P_2)\) is the total market demand at price \(P_2\)
  2. When \(P_1 > P_2\):
    • Firm 1 captures none of the market demand since consumers prefer the cheaper, identical product from Firm 2. Thus, \(Q_1 = 0\)
  3. When \(P_1 < P_2\):
    • Firm 1 captures the entire market demand because its price is lower. Hence, \(Q_1 = Q(P_1)\)

Residual Demand Bertrand

\[ Q_1 = \begin{cases} 0 ,& P_1>P_2 \\ \frac{1}{2}Q(P_2) ,& P_1=P_2 \\ Q(P_1) ,& P_1 < P_2 \end{cases} \]

Residual Demand Bertrand

Example (continued)

Bertrand Equilibrium

  • Definition: The Bertrand equilibrium is reached when each firm sets its price such that it maximizes its profits and neither firm can increase profits by unilaterally changing its price.

  • Reaching the Bertrand Equilibrium:

    • With identical products and rational, profit-maximizing firms, the equilibrium occurs when both firms set their prices equal to marginal cost (\(P = MC\)).
    • If one firm undercuts the other by even a small amount, it captures the entire market. Thus, to avoid losing the market, firms race to the bottom, stopping at \(P = MC\).

Bertrand Equilibrium

  • Characteristics:
    • The Bertrand equilibrium results in the competitive price level, even with only two firms in the market.
    • Total market supply meets all demand at this price level, as firms have sufficient capacity.

Bertrand vs. Other Market Structures

  • Bertrand vs. Cournot:
    • In Cournot competition, firms choose quantities, leading to higher prices than marginal costs due to the strategic reduction in total output.
    • In Bertrand competition, firms choose prices, leading to a price equal to marginal cost, similar to perfect competition outcomes, even with only two firms.
  • Bertrand and Perfect Competition:
    • Both result in prices equal to marginal costs; however, perfect competition assumes an infinite number of firms, while Bertrand demonstrates that two firms are sufficient to reach competitive pricing under price competition.

Implications for Market Outcomes

  • Cournot Model: Higher prices and lower quantities than in perfect competition, with profits for each firm.
  • Bertrand Model: Identical to perfect competition in terms of price and quantity, with no profits for each firm.